I have 2 systems running Debian at home and I am trying to minimize the bandwidth use while upgrading the packages. How can I increase the size limit of the cache, so that I could just copy the packages over LAN? (currently they get removed too quickly)

See /etc/cron.daily/apt for comments on available options, and the transition from APT::Archives::*. (Ubuntu 14.04 ships /etc/apt/apt.conf.d/20archive which sets APT::Archives::MaxSize. When they transition to APT::Periodic in that file, it will matter that you put your local changes in a file with a higher number.)

This sets the maximum size of the cache in MiB. If the cache
is bigger, cached package files are deleted until the size
requirement is met (the biggest packages will be deleted
first).

It's now APT::Archives::MaxSize, not ...Periodic.... (Ubuntu 14.04, apt 1.0.1ubuntu2.6) There's also ...::MinAge and ...::MaxAge (in days, I guess since the defaults are 2 and 30). I assume MinAge is a grace period for exceeding MaxSize.
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Peter CordesFeb 4 at 2:32

Also note that if either machine does apt-get autoclean, any package that doesn't show up in its list as available will be removed. `
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Peter CordesFeb 4 at 2:36

Also, Ubuntu ships /etc/apt/apt.conf.d/20archive, (dunno about Debian). Files are processed in order, with later ones able to override earlier ones. (I checked with apt-config dump.) You should put your changed in /etc/apt/apt.conf.d/99local-archive.
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Peter CordesFeb 4 at 3:58

hmm, I wonder if you can set Dir::Cache::archives to an absolute path (like /net/tesla/var/cache/apt/archives/), instead of the default relative path (archives/). Might be easier on a laptop that isn't always kept at home, easier than manually mounting over your local /var/cache/apt/archives/.
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Peter CordesFeb 4 at 4:05

Oops, ignore my first comment. lists.debian.org/deity/2014/03/msg00182.html and /etc/cron.daily/apt indicate that APT::Periodic is the new one, and Archives is the deprecated one. I guess the cron job is the only thing to ever prune the cache, since the size/age stuff is now specific to it.
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Peter CordesFeb 4 at 5:18

To set the cache size for apt, you need to edit apt.conf. You can get information on how to configure it in man apt.conf. If it is not already present create it in /etc/apt/apt.conf or you may need to edit /etc/apt/apt.conf.d/70debconf file, please refer specific version's documentation.

Debian provides Cache-Limit option in APT Group in apt.conf. You can experiment with it.

man apt.conf says,

Cache-Limit
APT uses a fixed size memory mapped cache file to store the
'available' information. This sets the size of that cache.